Definition:Group Automorphism

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Definition

Let $\struct {G, \circ}$ be a group.

Let $\phi: G \to G$ be a (group) isomorphism from $G$ to itself.


Then $\phi$ is a group automorphism.


Examples

Constant Product on Real Numbers

Let $\struct {\R, +}$ denote the real numbers under addition.

Let $\alpha \in \R$ be a real number.


Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \alpha x$

Then $f$ is a (group) automorphism if and only if $\alpha \ne 0$.


Also see

  • Results about group automorphisms can be found here.


Linguistic Note

The word automorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus automorphism means self structure.


Sources

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